Exercise 3, § 1

Let G be a monoid and E = K[G] its algebra (Algebra, Chapter III, § 2, no. 6). (a) Show that there exists on E one and only one cogebra structure such that c(g) = g \otimes g for all g \in G; this structure is compatible with the algebra structure on E and makes E into a cocommutative bigebra whose counit \varepsilon is such that \varepsilon(g) = 1 for all g \in G. (b) Show that every primitive element of E is zero. Deduce that, if G \neq \{e\}, the bigebra E admits no filtration compatible with its bigebra structure (apply Exercise 2). (c) Suppose that K is an integral domain. Show that the elements of G are the only non-zero elements x \in E such that c(x) = x \otimes x. Show that E has an inversion i (cf. Algebra, Chapter III, § 11, Exercise 4) if and only if G is a group and that in that case i(g) = g^{-1} for all g \in G.